The Power of Theorems

[neno89]

Hello! I need help solving this problem..i'm not sure how to do it. Can someone help me? Thanks!

The radius of each circle is 3. Triangle WXY is equilateral.
a. Find WY.
b. Find the ratio of the perimeters of triangle ABC, triangle PQR, and triangle WXY

 

[soroban]

Hello, neno89!

Part (a) is the tricky part . . .

The radius of each circle is 3. Triangle WXY is equilateral.
a. Find WY.
b. Find the ratio of the perimeters of triangle ABC, triangle PQR, and triangle WXY

Examine the bottom row of circles . . .

                  /                                       \
                 /  * * *           * * *           * * *  \
                /*         *     *         *     *         *\
               *             * *             * *             *
             /                                                 \
            / *       Q   r   *   r   C   r   *   r   R       * \
           /  *       * - - - * - - - * - - - * - - - *       *  \
          /   *       :       *               *               *   \
         /            :r                                           \
        /      *      :      * *             * *             *      \ 
       /         *    :    *     *         *     *         *         \
      *-------------*-*-*-----------*-*-*-----------*-*-*-------------*
      X               M - - - - -   4r  - - - - - - - N               Y

We want $\displaystyle \,WY\,=\,XY$

We see that: $\displaystyle \,MN\,=\,QR\,=\,4r$

We need only $\displaystyle \,XM\,=\,NY$


In right triangle $\displaystyle QMX:\;\angle QXM\,=\,30^o,\;QM\,=\,r$
$\displaystyle \;\;$Hence: $\displaystyle \,XM\,=\,\sqrt{3}r$

Therefore: $\displaystyle WY\:=\:XY\:=\:4r\,+\,2\sqrt{3}r\:=\:2(2\,+\,\sqrt{3})r$

$\displaystyle \;\;$I'll let you plug in $\displaystyle r\,=\,3$ . . . and solve part (b).